222 
Mr. Knight on the construction 
In like manner we might investigate approximations of the 
fourth and following orders : but this kind of research has 
very little use, and the Proposition was inserted for a different 
purpose. 
Prop. III. 
5- Supposing that , in the last Proposition , n=ri r =n" = &c. = — 1 . 
It is required to find the law of the converging expressions for 
L . x. 
In this case the four first transformations give 
*=(*—: Oxjrr 
*=(*_!) X--'x£=g 
v ' x—z 
X— I (x— l) (x — 3) x(x—z) s 
x=(x—i) X 
x=z(x— l) x 
X —2 (x—2) 1 X ( X l) 3 (x — 3) 
X 1 _ ( X 1 (x— 3) ^ (x — Q (x — 3)? ^ x(x — z } 6 (x — 4). 
X 2 x (x-Z y ' X (X— 2) X (^_l )4 
which, put into logarithms, give 
L.x^L^-U+Lf^) 
L . x=aL(x— 1)— L(x— 2)+L(ifc^) 
L . x= 3 L(x-i)-3L(x- 2 )+L(x-3)-fL( ( - ^-gl 3) ) 
L . x= 4 L(x-i)-6L(x- 2 )+4L(x^3)_L(x-4)+l(|^1^ 
where a coincidence may be observed between the coefficients 
and those in the binomial theorem ; and it is easily shown 
that the same coincidence will have place, how far soever 
we continue the method ; or that, in general, the converging 
expression will be 
L . x=4 L(X— 0 — ^L(x- 2 ) + ^ - -»- 2) L( x — 3) — ... 
n[n— 1 jt[n—i)(n — 2 )( n — 3) 
*x(a — ?■) 1,2 X(z— 4 ) 1>2 3 4 -f &c. 
n n(n — i)(n — 2) 
(af—i) 1 X{x— 3) 1,2,3 X&c. 
